Kalman-Filtering formulation details for Dynamic OD passenger matrix estimation

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1 Lecture Notes in Management Science (24) Vol. 6: t International Conference on Applied Operational Researc Proceedings Tadbir Operational Researc Group Ltd. All rigts reserved. ISSN 28-5 (Print) ISSN (Online) Kalman-Filtering formulation details for Dynamic OD passenger matrix estimation Lídia Montero Esteve Codina and Jaume Barceló Department of Statistics and Operations Researc Barcelona TECH-PC Carrer Jordi Girona Barcelona Spain {lidia.montero esteve.codina jaume.barcelo}@upc.edu Abstract. In tis paper we describe ow to estimate time-sliced origin-destination (OD) matrices for passenge in a public transport networ based on counts of ICT (Intelligent Communication Tecnology) devices carried by passenge at equipped transit-stops. Te transit assignment framewor is based on optimal strategy wic determines te subset of pats related to te optimal strategies between all OD pai for te wole orizon of study. Details are provided on ow to build te involved equations in a linear Kalman filtering model formulation wic is defined by te auto for a toy networ tat is proposed to validate te approac. Keywords: matrix estimation; alman-filtering; advanced transport information systems Introduction Te estimation of dynamic OD passenger matrices as received little attention in te literature due to te difficulties in collecting real-time passenger data. However new tecnologies ave generated proposals for automated passenger count (APC) and Automated Data Collection System (ADCS) applications to be used in transport planning wit a focus on te transit Origin-Destination inference. However OD matrices are not yet directly observable; consequently it as been natural to resort to indirect estimation metods. Wong and Tong (998) proposed a maximum entropy estimator tat employs te scedule-based approac for dynamic transit assignment. Ren (27) proposed a generalized least squares bilivel approac for estimating time-dependent passenger matrices in congested scedule-based transit netwo tat ave automatic passenger counts (APC) and prior OD matrices. Te proposal was tested in a toy networ but no recent wo from te autor ave confirmed eiter offline or online applicability to large scale transit netwo. Kostaos et al (2) proposed te use of passenge Bluetoot mobile devices to derive passenger OD matrices in a simplified context.

2 L Montero et al 83 Our aim is to explore te possibility of maing use of tis new data to estimate real-time dynamic Origin/Destination matrices of passenge in a transit networ using a sample of equipped passenge provided by data collected from antennas located in a subset of transit stops. Te remainder of tis paper is structured as follows. Fitly we describe te formulation for estimating passenger matrices in public transport. Next te approac is tested and validated on a toy networ. And finally te conclusions are stated. Model Formulation Notation is defined in Table. Some aspects of te data model and formulation statement tat ave to be considered are: Te demand matrix for te period of study is assumed to be divided into several time-slices accounting for different proportions of te total number of passenge in te time orizon. Te approac assumes an extended space state variable for M+ sequential time intervals of equal lengt t (between 5 and minutes for transit matrices) in order to consider non-instantaneous travel times. M sould guarantee traveing te networ. Bluetoot antennas are ICT senso assumed to be located at (some) transit-stops. OD pats involved in optimal strategies for transit trips in te period of study can be computed by any transportation planning software tat includes a strategy-based equilibrium transit assignment for istoric demand. We do not ave a strategy-based dynamic transit assignment tool available because we faced a delay in te development; so we ave used EMME4 (23) to define state-variables in our tests. To map te OD pats involved in optimal strategies we used te EMME output to build input files of te MatLab data model to systematically program in pyton tose pats involved in optimal strategies from centroid i to j and wic pass troug a pair (rs) of ICT senso. Table. Definition of model variables ~ Q i q~ i Q i ~ y q : q i : y q : ~ G ije G ije g ije g ~ : ije Historic total number of passenge and ICT equipped passenge accessing a transit unit at any stop inside te transportation area modeled by centroid i at time interval. Total number of passenge and BT equipped passenge accessing a transit unit at any stop inside te transportation area modeled by centroid i at time interval. Historic and actual number of equipped passenge crossing ICT sensor q or s from te pair of senso (rs)=q at time interval Total number of current G ije and istoric G ~ ije passenge as well as current g ije and istoric g ~ ije equipped passenge accessing centroid i at time interval and eaded towards j using pat e.

3 84 Lecture Notes in Management Science Vol. 6: ICAOR 24 Proceedings g ije : z z~ : u : u ije : t : State variables are deviates of equipped passenge accessing centroid i during interval and eaded towards centroid j using g g g ~. pat e wit respect to istoric data Te current and istoric measurements of equipped passenge during interval a column vector of dimension Q plus balance equations. Fraction of equipped passenge tat require time intervals to reac ICT sensor in transit stop s at time interval from ICT sensor r. Time-varying model paramete. Te values of u are updated according to te measurements of te ICT senso Fraction of equipped passenge detected at interval wose trip from centroid i to sensor s from sensor r migt use te OD pat e and tat last time intervals of lengt t M. Average measured travel time for equipped passenge captured by te pair of ICT senso (rs) crossing sensor s during interval ije ije ije Te total number of origin centroids (related to transportation zones) is I identified by index i from to I; te total number of ICT senso is P and te considered pai of ICT senso (rs) is Q were P ICT senso are located eiter at bus-stops or at segments in te inner networ; and te total number of pats (K) corresponding to optimal (static) transit strategies from te istoric OD transit matrix for te period of study. Eac equipped transit stop could be considered eiter an origin or a destination and it models a transit-stop tat migt be sared by several transit lines; but we prefer to estimate OD transit trips between OD pai not between transit-stops. Te state variables g ije () are assumed to be stocastic in nature. An autoregressive model of order r <<M is used to relate OD pat flow deviations at te current time to te OD pat flow deviations of previous time intervals. Te state equations are: Δg r DwΔg w w w were w() as zero mean wit a diagonal covariance matrix D w are transition matrices wic describe te effects of previous OD flows g ije (-w+) on current flows g ije (+) for w = r. If no convergence problems are detected we assume simple random wals in our researc in order to provide te most flexible framewor for state variables. Tus our fit trial will be r= and te D w matrix becomes te identity matrix. Te relationsip between te state variables and te measurements involves time-varying model paramete (congestion dependent since tey are updated from sample travel times provided by equipped passenge). Tis is applied using a linear transformation tat conside: W and Te number of equipped passenge fit detected in an equipped transit-stop r (lined/related to one or more origin zones i) during time intervals in... -M q r. H<M time-varying model paramete in te form of fraction matrices () u ije.

4 L Montero et al 85 Te H adaptive fractions tat approximate te travel time distribution between te pair of ICT senso (rs) notated asu and tat extend tou ije. Tese are updated from travel time measures provided by ICT senso. At time interval te values of te observations are determined by tose of te state variables at time intervals - -M: Δz F( ) g( ) v( ) (2) were v( ) is wite Gaussian noise wit covariance matrix R. F( ) maps te state vector g() onto te current blocs of measurements at time interval : counts of equipped passenge between a pair of ICT senso accounting for time lags and congestion effects and balances for origin zone preservation flow if available (for example one origin zone identified as te only source of passenge for a transit stop). Te solution sould provide estimations of te OD passenger matrices between OD pai for eac time interval up to te -t interval once observations of ICT equipped passenge at te bus-stops equipped wit wifi antennas up to te -t interval are available. KF prediction of OD trips for ICT-equipped passenge up to some intervals aead as to be considered and expanded according to istoric profiles (for day-type and time-period) in order to feed a dynamic transit assignment tool tat will provide te forecasted transit line loads boardings/aligtings at transit-stops in te sort-term future. Here we consider a 3 min forecasting orizon. Description of te Test Networ Te formulation to be detailed as been programmed as a MatLab prototype (named KFX3T). Proper coding as been verified wit a toy networ presented in Fig.. Te OD matrix consists of four non-zero OD transit flows (8) (9) (28) and (29) (identified as to 4). ICT senso are assumed to be available at nodes and 7 (sensor IDs are respectively to 4). Transit lines are L to L4: L from 3 to 6 L2 from 3 to 7 L3 from 4 to 6 and L4 from 4 to 7. Headway for lines L and L4 is 5 min and oterwise set to min. Flow is distributed at origins using a logit model wit scale parameter.2. Travel speed for all lines is 2m/ and boarding time is neglected to simplify te description of te example. Half eadway for eac line is incorporated into experienced travel times (travel impedance). In-veicle travel times result in.8 min for segments on L and L4 and 7.5 min for eac L2 and L3 segments. Te set of OD pats tat compose according to EMME (23) te optimal transit assignment strategies for te wole period of study are described in Table 2. Tus te number of OD pai is 4 te number of OD pats is and tere are 5 feasible pai of equipped transit stop captures (rs). We assume a subinterval of t=5 min for a our orizon of study.

86 Lecture Notes in Management Science Vol. 6: ICAOR 24 Proceedings OD 8 9 8 2 2 6 8 Fig..Test networ: lin and node identifie ICT senso (diamond nodes) and transit lines (L to L4) Table 2.

5 86 Lecture Notes in Management Science Vol. 6: ICAOR 24 Proceedings OD Fig..Test networ: lin and node identifie ICT senso (diamond nodes) and transit lines (L to L4) Table 2. Description of OD pats related to optimal strategies in transit assignment OD Pat/ Optimal Estrategy OD pat lines Table 3. Time-varying model paramete time from stop r in time intervals u OD Pat/ Optimal Estrategy ODPat id in OD pair OD pair ICT id OD pat lines ODPat id in OD pair OD pair ICT id / L =(8) 6/3 L 3=(28) 2/ L3 2 =(8) 23 7/3 L3 2 3=(28) 23 3/ L4 3 =(8) 24 8/3 L4 3 3=(28) 24 4/2 L2 2=(9) 34 9/4 L2 4=(29) 34 5/2 L4 2 2=(9) 24 /4 L4 2 4=(29) 24 : proportion of passenge arriving at stop s at s. t. u s. t. u s. t. u Orig TT(rs)(min) (ODpat id t 5' t 5' in for = ICT stop) t 5' i (rs) Any =2 =4 >>4 [3] (43) (93) 2 2 [4] (44) (94) [23] (23) (73) 2 2 (34)(84) 2 [24] (54)(4) u - [34] (44) (94) u 34 34

6 L Montero et al 87 Model Building for te Test Networ For eac [rs] pair of ICT senso u account for temporal dispeion of experienced travel times and tey are applied to all OD pats lining equipped transit stops (rs). We also assume tat te extended vector of states is te current one and tat M te number of t subintervals is set to M=5. Initially deviates of te OD pat flows are set equal to te istoric OD pat flows and % of equipped passenge is assumed for validation purposes; tus g ije. Time-varying model paramete u at initialization are set according to segment line speeds (2m/) and for > tey are defined as te average experienced travel impedances as indicated in Table 3. Ten if u for equal to ten tat would mean te a priori travel time to get to stop s from r for all te passenge is one time subinterval in te range ( t2 t] minutes or in oter words between 5 and min. Let g() be a column containing state variables for intervals M of dimension (M+)x=(5+)x=6. Te extended vector state as 6 components in te example and for r = te state variable equations are defined as a random wal wic can be expressed for te extended state vector as a sifting operator (Eq. ) by setting D as a 6x6 matrix wit non-null submatrix components I (identity matrix x). I I D I Te time-varying linear operator relates state variables and current observations for time interval in Equation (2). In te example z is a vector of dimension (Q+L)=5+=6 since ICT senso and 2 (located at nodes 3 and 4) do not clearly identify eac one of tem as te natural receptor of eiter origin centroid or 2. However by adding up teir captures te total flow generated by origin zones and 2 provides conservation flow equation. We refine te details of Eq. (2): Δz A() E ) T v () (3) v () Δg() F( ) g( ) v( ) ( 2 A discretization of te travel-time distributions between pai of equipped transit stops [rs] (5 pai) in H=2 bins is enoug in tis case (te fit bin is represented by a priori travel times). Ten if Ci is te number of OD pats originating in te networ at entry centroid i for i I ten C =5 and C 2 =5 in our case leading to te definition of one balance equation troug B and E() in Eq. (3) as:

7 88 Lecture Notes in Management Science Vol. 6: ICAOR 24 Proceedings C 5 B C2 5 E B A matrix in Equation (3) is composed by appending identity matrices of dimension 5 M+=6 times: A I Q I Q M and eac ( ) for = M models networ structure and travel time delays in terms of fractions on travel time bins (time-varying model paramete affecting state variables wose OD subpats are intercepted by some pai of ICT-equipped transit stops (rs) at s in interval ). Implementation guarantees tat implicit structural restrictions are satisfied (see Equation 4): u H u ( r s) H ( r s) uije ( ) ( ) Q u ije H uije ( r s) H ( r s) captured pat ije () and M (4) () is (+M) x (+M)Q=6*x6*5=6x3. In te example u ije ( ) > is te fraction of te equipped passenge tat were detected at equipped transit stop r ( intervals before ) and wo arrive at sensor s at interval (tis fraction applies to all captured OD pats). For = we define submatrices x5 in () since passenge tat enter te system during subinterval = cannot reac any equipped transit stop s from any detector at an equipped stop r (minimum time is 7.5 min). We sip a transit by applying =23 but once =4 ten all pai (rs) ave already received experienced travel times from passenge and approximations to travel time distributions are updated. For =4 we ave to average travel times in Table 3:

8 L Montero et al x And for >>M we would ave (always referring to Table 3 travel times): 3 2 x5 5 4 And te rigt-and side of Eq. (3) is fully described for te example. Te left-and side refe to deviates of counts for eac pair (rs) of ICT equipped transit stops (5 in te example) and for conservation flow equation (last row) for all OD pats entering te system at interval in Eq. (3). Tis concludes te model building details for te proposed formulation. Te standard report files related to OD flows nodes lins etc. in te EMME (23) model were fitted to woeet formats (fixed column.csv files) directly or by using pyton scripting. Fixed column.csv data files are read by KFX3T model building procedures. Te KFX3T internal data model is split into MAT files wic are loaded as needed into te program as: Global.mat Tuning.mat Grap.mat Demand.mat Measures.mat. Additionally two extra MAT files ave been included for internal use in order to simplify te access to some critical structures: AccDem.mat (OD pai and OD pats) and AccMes.mat (OD pats captured by eac defined sensor and pai of ICT senso).

9 9 Lecture Notes in Management Science Vol. 6: ICAOR 24 Proceedings Conclusions Model building for a KF formulation for estimating dynamic transit matrices as proposed by te auto as been detailed in a toy example. An EMME (23) model for te test networ was developed to feed te internal data model for te KFX3T prototype implemented in MATLAB. We developed a discrete event simulator to emulate passenger counts and travel times in te test networ. KFX3T as been validated and convergence olds but a fine tuning is needed. Sensibility to a priori OD flows per interval and covariance matrices for state variables and counts are being tested. Testing on medium-sized netwo is te next step to be undertaen in te immediate future. Acnowledgments Tis researc is funded by project TRA C3-2 of te Spanis R+D National Programs. References Antoniou C Ciuffo B Montero L Casas J Barceló J Cipriani E Djuic T Marzano V Nigro M Bullejos M Perarnau J Breen M Punzo V and Toledo T (24). A framewor for te bencmaring of OD estimation and prediction algoritms. Presented at 93rd TRB Annual Meeting Wasington D.C. Aso K and Ben-Aiva M (2). Alternative Approaces for Real-Time Estimation and Prediction of Time-Dependent Origin-Destination Flows. Transportation Science vol.34 no. pp Barceló J Montero L Bullejos M Linares M.P and Serc O (23). Robustness And Computational Efficiency Of A Kalman Filter Estimator of Time Dependent OD Matrices. Exploiting Traffic Measurements from Information and Communications Tecnologies. Transportation Researc Record pp Barceló J Montero L Bullejos M Serc O and Carmona C (23). A Kalman Filter Approac for te Estimation of Time Dependent OD Matrices Exploiting Bluetoot Traffic Data Collection. JITS -Journal of Intelligent Transportation Systems vol. 7(2) pp Cipriani E Gemma A Nigro M (23). A bi-level gradient approximation metod for dynamic traffic demand estimation: sensitivity analysis and adaptive approac. In: Proceedings of te 6t International IEEE Annual Conference on Intelligent Transportation Systems (ITSC 23) Te Hague Te Neterlands October 6-9. EMME 4..8 (23) ser s Manual. INRO Montreal ( Florian M and Hearn D (995). Networ Equilibrium Models and Algoritms. In: Capter 6 in M.O. Ball et al. Eds. Handboos in OR and MS Vol.8 Elsevier Science B.V. Kostaos V Camaco T and Mantero C (2). Wireless detection of end-to-end passenger trips on public transport buses. In proceedings of te IEEE Conference on Intelligent Transportation Systems Funcal Portugal pp Lin P and Cang G (27). A generalized model and solution algoritm for estimation of te dynamic freeway origin-destination matrix. Transportation Researc B vol.4 pp MatLab (22). MatWo Inc. ( Ren H.L (27). Origin-Destination Demands Estimation in Congested Dynamic Transit Netwo. In Proceeding of te 4t International Conference on Management Science and Engineering Harbin August Wong S.C and Tong C.O (998). Estimation of time-dependent Origin-Destination matrices for Transit netwo. Transportation Researc B vol.32 no. pp Elsevier Science Ltd. Wu J (997). A real-time origin-destination matrix updating algoritm for on-line applications. Transportation Researc B vol. 3-5 pp

Dynamic OD Matrix Estimation Exploiting Bluetooth Data in Urban Networks

Dynamic OD Matrix Estimation Exploiting Bluetooth Data in Urban Networks J.BARCELÓ L.MONTERO M.BULLEJOS O. SERCH and C. CARMONA Department of Statistics and Operations Research and CENIT (Center for Innovation